eigenclass.rb

class Foo
  def self.blah
    "foo"
  end
end

puts Foo.blah # "foo"

Foo.define_singleton_method :blah, do
  "hijink"
end

puts Foo.blah # "hijink"

eigenclass2.rb

class Foo
  def self.blah
    "foo"
  end
end

puts Foo.blah # "foo"

module FooExtend
  def blah
    "hijink"
  end
end

Foo.singleton_class.send :extend, FooExtend

puts Foo.blah # "foo"
puts Foo.singleton_class.blah # "hijink" 

eigenclass3.rb

class Foo
  def self.blah
    "foo"
  end
end

puts Foo.blah # "foo"

singleton_class = class << Foo; self; end;
def singleton_class.blah
  "hijink"
end

puts Foo.blah # "foo"
puts singleton_class.blah # "hijink"